The synchronization of remote systems has application in communications and related fields. Conventional synchronization theory applies to systems exhibiting linear behavior (stationary) or cyclic nonlinear behavior.
Nonlinear systems operating in the chaotic regime, that is, evolving chaotically, are characterized by extreme sensitivity to initial conditions and by broadband spectra. Extreme sensitivity to initial conditions means that two identical chaotic systems started under slightly different initial conditions quickly evolve to vastly different and uncorrelated states even though the overall patterns of behavior will remain the same. In bounded systems, the states do not diverge indefinitely, but repeatedly fold back. Lyapunov exponents measure such divergence and a system evolving chaotically is synonymous with a system having at least one positive Lyapunov exponent.
The evolution of a chaotic system is nonperiodic (there are no cycles of repetition whatsoever), becomes increasingly difficult to predict over time, and often has a complex, fractal character. Synchronization of remote chaotic systems has application in many communication related fields, such as broadband, spread-spectrum, multiplexing, security, pattern recognition, encryption, coding communications, in control devices relying on wide-frequency-band synchronized signals, phase locking, massively parallel systems, robotics, and physiology.
U.S. Pat. No. 5,245,660 to Pecora and Carroll teaches production of synchronized chaotic signals by a transmitter and receiver, the transmitter driving the receiver with a chaotic communications signal. The receiver includes a duplicate of a stable part of the transmitter. Currently co-pending U.S. application Ser. No. 08/129,495 by Pecora and Carroll extends this idea to cascaded nonlinear systems in which the receiver output signal is synchronized with the chaotic communications signal that drives the receiver. In such a cascaded system, a remote receiver has access to the chaotic communications drive signal. The overall system can be used for encryption of information in the chaotic communications drive signal and recovery of the information by the cascaded receiver.
If the transmitter and receiver are nonautonomous, that is, forced by externally provided periodic signals, then the external forcing signals might have differing phase and frequency, thus making it impossible to synchronize such systems by traditional phase locking techniques.